$a+b+1=0$ is the condition for the quadratic equations ${ x }^{ 2 }+ax+b=0$ and ${ x }^{ 2 }+bx+a=0$ to have a common root.

Let the common root be r. Substituting r into both x^2+ax+b=0 and x^2+bx+a=0 yields r^2+ar+b=0 and r^2+br+a=0. Subtracting the second equation from the first gives (a-b)r+(b-a)=0, which factors to (a-b)(r-1)=0. If a does not equal b, the common root r must be 1. Substituting r=1 into the first equation gives 1+a+b=0, which is exactly a+b+1=0. This confirms the statement is true.

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